Study of Scheme Transformations to Remove Higher-Loop Terms in the $\beta$ Function of a Gauge Theory

TL;DR

This paper investigates scheme transformations that remove higher-loop terms in the beta function of gauge theories, analyzing their applicability at infrared fixed points and their impact on different gauge theories.

Contribution

It introduces and studies the $S_{R,m}$ scheme transformation for removing higher-loop terms, extending previous work and analyzing its applicability in SU(N_c) and U(1) gauge theories.

Findings

The $S_{R,3}$ transformation is applicable only within a limited range of fermion numbers $N_f$.

Higher-loop terms influence the beta function significantly at small fixed-point couplings.

The study provides explicit formulas for transformed beta function coefficients up to eight loops.

Abstract

Since three-loop and higher-loop terms in the $\beta$ function of a gauge theory are scheme-dependent, one can, at least for sufficiently small coupling, carry out a scheme transformation that removes these terms. A basic question concerns the extent to which this can be done at an infrared fixed point of an asymptotically free gauge theory. This is important for quantitative analyses of the scheme dependence of such a fixed point. Here we study a scheme transformation $S_{R,m}$ with $m \ge 2$ that is constructed so as to remove the terms in the beta function at loop order $\ell=3$ to $\ell=m+1$, inclusive. Starting from an arbitrary initial scheme, we present general expressions for the coefficients of terms of loop order $\ell$ in the beta function in the transformed scheme from $\ell=m+2$ up to $\ell=8$. Extending a previous study of $S_{R,2}$, we investigate the range of applicability of the $S_{R,3}$ scheme transformation in an asymptotically free SU($N_c$) gauge theory with an infrared zero in $\beta$ depending on the number, $N_f$, of fermions in the theory. We show that this $S_{R,3}$ scheme transformation can only be applied self-consistently in a restricted range of $N_f$ with a correspondingly small value of infrared fixed-point coupling. We also study the effect of higher-loop terms on the beta function of a U(1) gauge theory.